Further results on $[k]$-Roman domination on cylindrical grids $C_m \Box P_n$

Simon Brezovnik; Janez Žerovnik

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arXiv · 2603.25191

Further results on $[k]$-Roman domination on cylindrical grids $C_m \Box P_n$

Simon Brezovnik·Janez Žerovnik

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Abstract

In this paper, we study the $[k]$-Roman domination number of cylindrical graphs $C_m \Box P_n$. Our analysis begins with a general lower bound based on local neighborhood constraints, showing that $γ_{[k]R}(C_m\Box P_n) > (k+1)\left\lceil\frac{mn}{5}\right\rceil.$ By exploiting the connection between $[k]$-Roman domination and efficient domination, we characterize those cylindrical graphs whose optimal $[k]$-Roman domination number is realized by configurations with minimum possible local neighborhood weight. For fixed small values $m\in\{5,\ldots,8\}$, we construct explicit periodic $[k]$-Roman dominating functions that yield constructive upper bounds. These constructions are further refined using ceiling-type adjustments and reductions based on packing sets. A systematic comparison of the resulting bounds shows how their relative strength depends on the parameter $k$ and on the length of the path.

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Simon Brezovnik, Janez Žerovnik. 2026-03-26. Further results on $[k]$-Roman domination on cylindrical grids $C_m \Box P_n$. https://arxiv.org/abs/2603.25191

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